How to rotate set of textures using Texture Coordinate and Mapping nodes?
When you rotate a normal map, a good way of thinking about is that you are rotating the light direction as well. (Which is what you're seeing.)
To rotate correctly by 90°, you'll have to rotate the &quot;light direction&quot; back by -90°.
You can do this swapping the X and Y vectors. You also need to invert the (orig) Y. See here:

This is how you rotate by 180°:

EDIT: Re. angles which aren't multiples of 90°: It is possible, but a bit more complicated.
Here's a node setup:

So, I made a Node Group. Let me know if this works correctly or if I made a mistake. ;)

EDIT2: scurest has added an easier method in the comments: Rotate the output of the Normal Map node via Mapping Node.
So, these two are identical: